# F77SB Introduction to Statistical Science B

Course co-ordinator(s): Prof Jennie C Hansen (Edinburgh).

Aims:

The aims of this course are

• To develop discrete probability models for data
• To understand important features of these models

Summary:

This course provides an introduction to some of the probability models for inference. The main topics covered are:

• Probability models: sample spaces and events; probability measures and their properties; simple probability calculations.
• Conditioning and Independence: conditional probability; `Chain Rule’ for computing probabilities; Bayes’ Theorem; Partition Theorem; independence of events; probability calculations using concepts of conditioning and independence.
• Random variables and their distributions, including the Binomial, Geometric, Poisson, Hypergeometric, and Indicator variables.
• Expectation and standard deviation of a random variable: definitions and properties of expectation and standard deviation.

## Detailed Information

Course Description: Link to Official Course Descriptor.

Pre-requisites: none.

Location: Edinburgh.

Semester: 2.

Syllabus:

1. Introduction to discrete probability models

• Models for random experiments:
• Sample spaces: events, set theory, Venn diagrams, de Morgan’s laws
• Probability functions and the axioms of probability: Choice of probability function, consequences of the axioms.
• Conditional probability and independence of events:
• Definitions of P(A|B) and the definition of independence
• `Chain rule’ for P(A intersection B)
• Bayes Theorem
• Partition Theorem and `tree diagrams’.

2. Special probability models for certain random experiments

• Simple equally likely models
• Sampling without replacement from a finite population:
• ordered samples: model assumptions, calculations and relevant counting arguments
• unordered samples: model assumptions, calculations and relevant counting arguments
• Building models for a sequence of independent sub
• experiments:
• General model assumptions
• Sampling with replacement: calculations and relevant counting arguments
• Bernoulli trials and Binomial models: calculations and relevant counting arguments
• Geometric models
• Other examples

3. Discrete random variables

• Definition of a discrete random variable
• Probability mass functions for discrete random variables
• Expectation and the properties of the expectation of a random variable.
• Variance and the properties of the variance of a random variable.
• Indicator variables

4. Special examples of discrete random variables.

• Binomial and Bernoulli variables
• Geometric and Negative Binomial variables
• Poisson variables
• Hypergeometric variables
• Indicator variables

Learning Outcomes: Subject Mastery

At the end of this course, students should be able to

• Carry out probability calculations for basic discrete probability models using standard techniques including the Partition Theorem, Bayes’ Theorem, and the Chain Rule.
• Work out the distribution, expectation, and standard deviation for a discrete random variable that describes the outcome of a random experiment.
• Identify an appropriate standard random variable to describe the outcome of a random experiment.
• Use the properties of expectation and indicator variables to work out the expected value of a counting variable.

Reading list:

A First Course in Probability by S.M.Ross, 7th ed. (Pearson 2006).

Assessment Methods:

2-hour final exam in May (80%) and continuous assessment (2 assignments) (20%)

SCQF Level: 7.

Credits: 15.

## Other Information

Help: If you have any problems or questions regarding the course, you are encouraged to contact the lecturer

VISION: further information and course materials are available on VISION